Two inverse problems of finding a coefficient in a parabolic equation

Камынин, В. Л.; Костин, А. Б.

doi:10.1134/s0012266110030079

Cited by 23 publications

(28 citation statements)

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“…Note that, under conditions (A 1 ), for any r ∈ E + , the solution of direct problem ( and the function r(x) ≥ 0 satisfies the estimate r( can be found in [9,11].…”

Section: Inverse Problem With Integral Observationmentioning

confidence: 99%

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Recovery of the coefficient of u t in the heat equation from a condition of nonlocal observation in time

Костин

2015

Comput. Math. and Math. Phys.

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The inverse problem of finding the coefficient ρ(x) = ρ 0 + r(x) multiplying u t in the heat equation is studied. The unknown function r(x) 0 is sought in the class of bounded functions, and ρ 0 is a given positive constant. In addition to the initial and boundary conditions (data of the direct problem), a nonlocal observation condition is specified in the form u(x, t)dμ(t) = χ(x) with a given measure dμ(t) and a function χ(x). The case of integral observation (i.e., dμ(t) = ω(t)dt) is considered separately. Sufficient conditions for the existence and uniqueness of a solution to the inverse problem are obtained in the form of easy to check inequalities. Examples of inverse problems are given for which the assumptions of the theorems proved in this work are satisfied.Keywords: coefficient inverse problems, inverse problem for the heat equation, nonlocal observation (or overdetermination) condition, sufficient conditions for the existence and uniqueness of a solution. ≥ 0 T ∫ COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS Vol. 55 No. 1 2015 KOSTINwhere μ(t) and χ(x) are given functions. Observation condition (1.2) is nonlocal in time t, and it was pro posed for a linear inverse problem in [19]. More specifically, it includes the final observation and the integral observation .If, for example, ω(t) = 1/T, then we have the condition of the integral mean of the function u(x, t):.Another typical example of a condition of form (1.2) is , where α k > 0 and t k are given numbers. This condition can be interpreted as setting a weighted mean value of u(x, t).The problem of determining the coefficient of u t in a parabolic equation from a final observation con dition was studied in [3,8]. The solution was considered in Hölder classes, which ruled out the possibility of discontinuous coefficients. In [6, 7] a similar coefficient inverse problem with final and integral obser vations was investigated in Sobolev classes. In [11] the coefficient of Δu was recovered in the case of inte gral observation. Note that the problem we consider below is not reduced to this case. In this paper, the results of [6,7] are extended to the more general overdetermination condition (1.2) and an iterative method for finding the coefficient r(x) in the class of bounded functions is proposed and justified. The case of integral observation for a smooth weight function ω(t), which has not been addressed earlier for such a problem, is considered separately. As a result, a unique solvability theorem for the inverse problem is proved under weaker constraints on the given functions. Note also that the inverse problem of recovering r(x) with a general observation of form (1.2), where μ(t) has a bounded variation on [0, T], has not been studied previously. However, even in the case of integral and final observations, Theorem 1 and 2 provide new (in comparison with [6-8]) unique solvability conditions for the inverse problem. Specifically, in contrast to [6][7][8], the initial data are no longer required to be zero, and an explicit estimate for the desir...

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“…Note that, under conditions (A 1 ), for any r ∈ E + , the solution of direct problem ( and the function r(x) ≥ 0 satisfies the estimate r( can be found in [9,11].…”

Section: Inverse Problem With Integral Observationmentioning

confidence: 99%

“…Inverse problems of finding coefficients in parabolic equations have been extensively studied (see, e.g., [1][2][3][4][5][6][7][8][9][10][11][12][13]). These problems are nonlinear, since the unknown functions enter into the equation in the form of products.…”

Section: Introductionmentioning

confidence: 99%

Recovery of the coefficient of u t in the heat equation from a condition of nonlocal observation in time

Костин

2015

Comput. Math. and Math. Phys.

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“…Nonlocal problems with conditions of such a type for other equations were considered in [7-9]. The investigation of problems with nonlocal with respect to time integral conditions, whose results are presented in this paper, show that the dimension of a domain, in which one finds a solution, is important, and that the solvability conditions can connect both the domain dimensions and the constraints on other initial data.Problems with nonlocal with respect to time conditions are tightly connected with inverse problems, whose condition of redefinition is integral one [10][11][12]. Thus defined conditions can be considered as a model of action of some apparatus, which registers physical fields [13].…”

mentioning

confidence: 99%

“…Thus defined conditions can be considered as a model of action of some apparatus, which registers physical fields [13]. As an example we adduce the condition of redefinition in the problem of definition of coefficient of parabolic equation [12] T 0 ω(τ )u(x, τ )dτ = χ(x).One of investigation stages of resolvability of the inverse problem is often the solution to the direct problem, and in addition it is not excluded that the direct problem can be nonlocal. This reasoning has forced the purely theoretical interest to the investigation of nonlocal problems with integral conditions and has led to the definition of problem with nonlocal initial conditions for the one-dimensional hyperbolic equation.…”

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A problem with nonlocal initial data for one-dimensional hyperbolic equation

Кириченко

Пулькина²

2014

Russ Math.

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In this paper we consider a boundary-value problem for one-dimensional hyperbolic equation with nonlocal initial data in integral form. We prove the existence and uniqueness of the generalized solution.Introduction. At the present time, problems with nonlocal conditions for partial differential equations are actively investigated. This interest is initiated by the necessity to generalize classical problems of Mathematical Physics in connection with the mathematical modelling of a series of physical processes, which are investigated by the modern natural sciences [1]. In those cases when the boundary of domain of physical process is not accessible for the immediate measuring, nonlocal conditions in the integral form can serve as additional information, which is sufficient for the unique resolvability of the corresponding mathematical problem. The present paper is dedicated to the investigation of solvability of a nonlocal problem with integral conditions. An essential part of publications, starting from the G. R. Cannon paper [2], which are dedicated to problems with integral conditions, contains investigations of parabolic equations. Problems with integral conditions for hyperbolic equations are considered in [3-6] and in references therein. We note that in the majority of publications, which are dedicated to problems with nonlocal integral conditions, one considers spatially nonlocal conditions. In the present paper we consider the problem with nonlocal with respect to time integral conditions for a hyperbolic equation. Nonlocal problems with conditions of such a type for other equations were considered in [7-9]. The investigation of problems with nonlocal with respect to time integral conditions, whose results are presented in this paper, show that the dimension of a domain, in which one finds a solution, is important, and that the solvability conditions can connect both the domain dimensions and the constraints on other initial data.Problems with nonlocal with respect to time conditions are tightly connected with inverse problems, whose condition of redefinition is integral one [10][11][12]. Thus defined conditions can be considered as a model of action of some apparatus, which registers physical fields [13]. As an example we adduce the condition of redefinition in the problem of definition of coefficient of parabolic equation [12] T 0 ω(τ )u(x, τ )dτ = χ(x).One of investigation stages of resolvability of the inverse problem is often the solution to the direct problem, and in addition it is not excluded that the direct problem can be nonlocal. This reasoning has forced the purely theoretical interest to the investigation of nonlocal problems with integral conditions and has led to the definition of problem with nonlocal initial conditions for the one-dimensional hyperbolic equation. *

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A numerical solution of an inverse diffusion problem based on operational matrices of orthonormal polynomials

Rashedi

2021

Math Methods in App Sciences

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The inverse problem of identifying the diffusion coefficient in the one‐dimensional parabolic heat equation is studied. We assume that the information of Dirichlet boundary conditions along with an integral overdetermination condition is available. By applying the given assumptions, the problem is reformulated as a nonclassical parabolic equation along with the initial and boundary conditions. Then we employ the direct technique based on the operational matrices for integration, differentiation, and the product of the orthonormal polynomials together with the Ritz–Galerkin technique to reduce the main problem to the solution of a system of nonlinear algebraic equations. Numerical simulations are presented to show the applicability of the proposed method.

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Two inverse problems of finding a coefficient in a parabolic equation

Cited by 23 publications

References 5 publications

Recovery of the coefficient of u t in the heat equation from a condition of nonlocal observation in time

Recovery of the coefficient of u t in the heat equation from a condition of nonlocal observation in time

A problem with nonlocal initial data for one-dimensional hyperbolic equation

A numerical solution of an inverse diffusion problem based on operational matrices of orthonormal polynomials

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